Evaluate $\log_{\sqrt{6}} (216\sqrt{6})$.
Answer: Let $x= \log_{\sqrt{6}}(216\sqrt{6})$.  Putting this in exponential notation gives $(\sqrt{6})^x = 216\sqrt{6}$.  Writing both sides with $6$ as the base gives us $6^{\frac{x}{2}} = 6^3\cdot 6^{\frac{1}{2}} = 6^{\frac{7}{2}}$, so $x/2=7/2$.  Therefore, $x=\boxed{7}$.